Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $r \neq 0$. $t = \dfrac{5r}{6r - 27} \div \dfrac{4r}{12r - 54} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $t = \dfrac{5r}{6r - 27} \times \dfrac{12r - 54}{4r} $ When multiplying fractions, we multiply the numerators and the denominators. $t = \dfrac{ 5r \times (12r - 54) } { (6r - 27) \times 4r } $ $ t = \dfrac {5r \times 6(2r - 9)} {4r \times 3(2r - 9)} $ $ t = \dfrac{30r(2r - 9)}{12r(2r - 9)} $ We can cancel the $2r - 9$ so long as $2r - 9 \neq 0$ Therefore $r \neq \dfrac{9}{2}$ $t = \dfrac{30r \cancel{(2r - 9})}{12r \cancel{(2r - 9)}} = \dfrac{30r}{12r} = \dfrac{5}{2} $